SRC/dtgevc.f(3)

📦

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💖 SRC/dtgevc.f(3)

okay so bestie, this is the Library Functions Manual page for SRC/dtgevc.f(3), and it is lowkey giving “linear algebra main character” energy. it’s a LAPACK routine page, so grab a lil snack and let’s get into it 💅✨

🎀 NAME

SRC/dtgevc.f — that’s the name, bestie, short and iconic.

✨ SYNOPSIS

🌈 Functions/Subroutines

subroutine dtgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl,
    vr, ldvr, mm, m, work, info)
    DTGEVC

🌟 Function/Subroutine Documentation

💫 subroutine dtgevc

subroutine dtgevc (character side, character howmny, logical, dimension( * )
    select, integer n, double precision, dimension( lds, * ) s, integer lds,
    double precision, dimension( ldp, * ) p, integer ldp, double precision,
    dimension( ldvl, * ) vl, integer ldvl, double precision, dimension(
    ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision,
    dimension( * ) work, integer info)
DTGEVC

🦋 Purpose

okay so here’s the vibe: DTGEVC computes some or all of the right and/or left eigenvectors of a pair of real matrices (S,P), where S is a quasi-triangular matrix and P is upper triangular. matrix pairs of this type are produced by the generalized Schur factorization of a matrix pair (A,B):

A = Q*S*Z**T,  B = Q*P*Z**T

as computed by DGGHRD + DHGEQZ. slay.

the right eigenvector x and the left eigenvector y of (S,P) corresponding to an eigenvalue w are defined by:

S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

where y**H denotes the conjugate transpose of y. the eigenvalues are not input to this routine, but are computed directly from the diagonal blocks of S and P. no cap, they just show up from the diagonal blocks 💫

this routine returns the matrices X and/or Y of right and left eigenvectors of (S,P), or the products Z*X and/or Q*Y, where Z and Q are input matrices. if Q and Z are the orthogonal factors from the generalized Schur factorization of a matrix pair (A,B), then Z*X and Q*Y are the matrices of right and left eigenvectors of (A,B). periodt 🫶

🌸 Parameters

SIDE — SIDE is CHARACTER*1. = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. 💅
HOWMNY — HOWMNY is CHARACTER*1. = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, specified by the logical array SELECT. ✨
SELECT — SELECT is LOGICAL array, dimension (N). If HOWMNY='S', SELECT specifies the eigenvectors to be computed. If w(j) is a real eigenvalue, the corresponding real eigenvector is computed if SELECT(j) is .TRUE.. If w(j) and w(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to .FALSE.. Not referenced if HOWMNY = 'A' or 'B'. 🦋
N — N is INTEGER. The order of the matrices S and P. N >= 0. 💖
S — S is DOUBLE PRECISION array, dimension (LDS,N). The upper quasi-triangular matrix S from a generalized Schur factorization, as computed by DHGEQZ. 🌟
LDS — LDS is INTEGER. The leading dimension of array S. LDS >= max(1,N). ✨
P — P is DOUBLE PRECISION array, dimension (LDP,N). The upper triangular matrix P from a generalized Schur factorization, as computed by DHGEQZ. 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S must be in positive diagonal form. 👑
LDP — LDP is INTEGER. The leading dimension of array P. LDP >= max(1,N). 💜
VL — VL is DOUBLE PRECISION array, dimension (LDVL,MM). On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the orthogonal matrix Q of left Schur vectors returned by DHGEQZ). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P); if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of (S,P) specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. Not referenced if SIDE = 'R'. 🌈
LDVL — LDVL is INTEGER. The leading dimension of array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N. 🎀
VR — VR is DOUBLE PRECISION array, dimension (LDVR,MM). On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Z (usually the orthogonal matrix Z of right Schur vectors returned by DHGEQZ). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P); if HOWMNY = 'B' or 'b', the matrix Z*X; if HOWMNY = 'S' or 's', the right eigenvectors of (S,P) specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. Not referenced if SIDE = 'L'. 💫
LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N. 🔥
MM — MM is INTEGER. The number of columns in the arrays VL and/or VR. MM >= M. 🌸
M — M is INTEGER. The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns. 💖
WORK — WORK is DOUBLE PRECISION array, dimension (6*N). ⭐
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex eigenvalue. 💀

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔥 Further Details

allocation of workspace, bestie — the receipts are here:

Allocation of workspace:
---------- -- ---------

   WORK( j ) = 1-norm of j-th column of A, above the diagonal
   WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
   WORK( 2*N+1:3*N ) = real part of eigenvector
   WORK( 3*N+1:4*N ) = imaginary part of eigenvector
   WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
   WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

Rowwise vs. columnwise solution methods: ------- -- ---------- -------- -------

finding a generalized eigenvector consists basically of solving the singular triangular system:

(A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

consider finding the i-th right eigenvector (assume all eigenvalues are real). the equation to be solved is:

      n                   i
 0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
     k=j                 k=j

where C = (A - w B) (the components v(i+1:n) are 0). 💅

the method is:

(1)  v(i) := 1
for j = i-1,. . .,1:
                        i
    (2) compute  s = - sum C(j,k) v(k)   and
                      k=j+1

    (3) v(j) := s / C(j,j)

step 2 is sometimes called the step, since it is an inner product between the j-th row and the portion of the eigenvector that has been computed so far. 🦋

the method consists basically in doing the sums for all the rows in parallel. as each v(j) is computed, the contribution of v(j) times the j-th column of C is added to the partial sums. since FORTRAN arrays are stored columnwise, this has the advantage that at each step, the elements of C that are accessed are adjacent to one another, whereas with the rowwise method, the elements accessed at a step are spaced LDS (and LDP) words apart. ✨

when finding left eigenvectors, the matrix in question is the transpose of the one in storage, so the rowwise method then actually accesses columns of A and B at each step, and so is the preferred method. 👑

Definition at line 293 of file dtgevc.f.

👑 Author (the generated-by one)

Generated automatically by Doxygen for LAPACK from the source code. 🫶

okay that’s the whole page, bestie! source/version: LAPACK Version 3.12.0; page: SRC/dtgevc.f(3); date: not listed in the footer, so that’s the whole footer vibe. mwah 💋✨🌸

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms